Date of Project

3-28-2025

Document Type

Honors Thesis

School Name

College of Arts and Sciences

Department

Mathematics

Major Advisor

Dr. Gregory Kelsey

Second Advisor

Dr. Susan White

Abstract

In a real vector space, a symmetric convex body containing the origin and compact under the Euclidean norm uniquely defines a norm via the Minkowski functional, where the closed unit ball of this norm is the convex body. This establishes a one-to-one correspondence between convex bodies and norms. The norm derived from the convex body allows for the definition of a metric, enabling the study of isometries within the space. However, the symmetries of the convex body itself do not completely determine the isometries of the associated Minkowski space. Instead, the symmetries of the most symmetric shape a convex body can be linearly deformed into correspond to Euclidean and possibly newly defined non-Euclidean isometries of the space. This thesis provides an accessible review of the connection between convex bodies and norms via the Minkowski functional and explores how the symmetries of convex bodies help predict the Euclidean and non-Euclidean isometries of the associated Minkowski space.

Share

COinS